TITLE: What is the standard method to estimate probabilities of events based on observations?
QUESTION [5 upvotes]: I start from a simple example. I through a (possibly unfair) coin 20 times. I got "eagle" and "tail" 15 and 5 times, respectively. Now I need to estimate probabilities for eagle and tail. For eagle I get 15/20 = 0.75 and for the tail I get 0.25.
However, I know that the probability that I got are not accurate. It is still possible that I have a fair coin (with probability for eagle equal to 0.5) and I get more eagles in my experiment just by chance.
Now I want to estimate probabilities of probabilities. In other words I want to know how likely it is that probability for eagle is 0.5 (or 0.3, or 0.9). I can solve this problem, but I would like to know if there is a name for this problem. I an also interested in the generalization of the problem for the case of more than two events (not just "eagle" and "tail").

REPLY [2 votes]: We want to find $p,0\le p\le 1,$ such that $${20\choose 15}p^{15}(1-p)^5$$ is maximum.
By simple calculus, this $p$ turns out to be 15/20, as expected.